Numerical Solution of Uncertain Beam Equations Using Double Parametric Form of Fuzzy Numbers

Abstract

Present paper proposes a new technique to solve uncertain beam equation using double parametric form of fuzzy numbers. Uncertainties appearing in the initial conditions are taken in terms of triangular fuzzy number. Using the single parametric form, the fuzzy beam equation is converted first to an interval-based fuzzy differential equation. Next, this differential equation is transformed to crisp form by applying double parametric form of fuzzy number. Finally, the same is solved by homotopy perturbation method (HPM) to obtain the uncertain responses subject to unit step and impulse loads. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the methodology.

Double Parametric Fuzzy Numbers Approximate Scheme for Solving One-Dimensional Fuzzy Heat-Like and Wave-Like Equations

“…the upper bound of fuzzy number. For the purpose of this paper, we only provide the definition of double parametric form of fuzzy numbers and its properties, while for the single parametric form, the reader can refer to [26] for further details. 2.2.…”

“…2.2. Double parametric form of fuzzy numbers [26]: Letμ = µ(r), µ(r) be a parametric form of fuzzy numberR; then one may represent the double parametric form in crisp values asμ (r, b) = b µ(r) − µ(r) + µ(r), where r, b ∈ [0, 1]. The embedding parameter b denotes the deforming parameter such that if b = 0 thenμ (r, 0) = µ(r) (lower bound fuzzy number), and if b = 1 thenμ (r, 1) = µ(r) (upper bound fuzzy number).…”

“…However, with the existing single parametric fuzzy approach of approximate-analytical methods, it is known that the analysis and computational work is complex, where an n x n completely fuzzy system has to be transformed into a much larger 2 n × 2 n crisp system. To alleviate this complexity issue, the double parametric approach, which is not only more general and simple but also requires less analysis and computational work, has been developed in studies on FPDEs involving beam equations and fuzzy delay differential equations [26,27]. The incorporation of additional embedding parameter that functions as a deforming parameter annihilates the necessity to transform the n × n completely fuzzy system to a larger system in the solving process but instead enables practitioners to maintain the same n × n crisp system order after transformation, which, in turn, reduces the entire work complexity.…”

Homotopy Perturbation Method for Solving Linear Fuzzy Delay Differential Equations Using Double Parametric Approach

“…On the other hand, for the double parametric form, the n x n fully fuzzy system is converted to the same order of crisp system, hence requiring a less amount of computation. The double parametric form, which has been employed in fuzzy differential equation, is more general and straightforward [16].…”